Abstract
The analysis of experimental random walks aims at identifying the process(es) that generate(s) them. It is in general a difficult task, because statistical dispersion within an experimental set of random walks is a complex combination of the stochastic nature of the generating process, and the possibility to have more than one simple process. In this paper, we study by numerical simulations how the statistical distribution of various geometric descriptors such as the second, third and fourth order moments of two-dimensional random walks depends on the stochastic process that generates that set. From these observations, we derive a method to classify complex sets of random walks, and resolve the generating process(es) by the systematic comparison of experimental moment distributions with those numerically obtained for candidate processes. In particular, various processes such as Brownian diffusion combined with convection, noise, confinement, anisotropy, or intermittency, can be resolved by using high order moment distributions. In addition, finite-size effects are observed that are useful for treating short random walks. As an illustration, we describe how the present method can be used to study the motile behavior of epithelial microvilli. The present work should be of interest in biology for all possible types of single particle tracking experiments.
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Abbreviations
- cdf:
-
Cumulative distribution function
- MSD:
-
Mean Square Displacement
- SPT:
-
Single Particle Tracking
References
Amblard, F., Maggs, A.C., Yurke, B., Pargellis, A., Leibler, S., 1996. Subdiffusion and anomalous local viscoelasticity in actin networks. Phys. Rev. Lett. 77(21), 4470–4473.
Bentil, D.E., 1998. Distribution of attachment events relative to actin binding sites as evidenced in a bidirectional actomyosin interaction model. Bull. Math. Biol. 60(5), 973–995.
Bertin, E.M., Bouchaud, J.P., 2003. Subdiffusion and localization in the one-dimensional trap model. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 67(2 Pt 2), 026128.
Borgdorff, A., Choquet, D., 2002. Regulation of AMPA receptor lateral movements. Nature 417, 649–653.
Block, S.M., 1998. Kinesin: what gives? Cell 93(1), 5–8.
Caspi, A., Granek, R., Elbaum, M., 2002. Diffusion and directed motion in cellular transport. Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 66(1 Pt 1), 011916.
Cherry, R.J., Georgiou, G.N., Morrison, I.E., 1994. New insights into the structure of cell membranes from single particle tracking experiments. Biochem. Soc. Trans. 22(3), 781–784.
Cherry, R.J., Smith, P.R., Morrison, I.E., Fernandez, N., 1998. Mobility of cell surface receptors: a re-evaluation. FEBS Lett. 430(1–2), 88–91.
Cognet, L., Tardin, C., Boyer, D., Choquet, D., Tamarat, P., Lounis, B., 2003. Single metallic nanoparticle imaging for protein detection in cells. Proc. Natl. Acad. Sci. USA 100(20), 11350–11355.
Coscoy, S., Waharte, F., Gautreau, A., Martin, M., Louvard, D., Mangeat, P., Arpin, M., Amblard, F., 2002. Molecular analysis of microscopic ezrin dynamics by two-photon FRAP. Proc. Natl. Acad. Sci. USA 99(20), 12813–12818.
Coudrier, E., Amblard, F., Zimmer, C., Roux, P., Olivo-Marin, J.C., Rigothier, M.C., Guillen, N., 2005. Myosin II and the Gal-GalNAc lectin play a crucial role in tissue invasion by Entamoeba histolytica. Cell Microbiol. 7(1), 19–27.
Craig, A.M., Lichtman, J.W., 2001. Getting a bead on receptor movements. Nat. Neurosci. 4(3), 219–220.
Dahan, M., Levi, S., Luccardini, C., Rostaing, P., Riveau, B., Triller, A., 2003. Diffusion dynamics of glycine receptors revealed by single-quantum dot tracking. Science 302(5644), 442–445.
Daumas, F., Destainville, N., Millot, C., Lopez, A., Dean, D., Salome, L., 2003. Confined diffusion without fences of a G-protein-coupled receptor as revealed by single particle tracking. Biophys. J. 84, 356–366.
Friedl, P., Wolf, K., 2003. Tumour-cell invasion and migration: diversity and escape mechanisms. Nat. Rev. Cancer 3(5), 362–374.
Flyvbjerg, H., Peterson, H.G., 1989. Error estimates on averages of correlated data. J. Chem. Phys. 91, 461–466.
Geerts, H., De Brabander, M., Nuydens, R., Geuens, S., Moeremans, M., De Mey, J., Hollenbeck, P., 1987. Nanovid tracking: a new automatic method for the study of mobility in living cells based on colloidal gold and video microscopy. Biophys. J. 52(5), 775–782.
Goulian, M., Simon, S.M., 2000. Tracking single particles within cells. Biophys. J. 79, 2188–2198.
Gross, S.P., Welte, M.A., Block, S.M., Wieschaus, E.F., 2000. Dynein-mediated cargo transport in vivo: a switch controls travel distances. J. Cell Biol. 148, 945–956.
Hughes, B.D., 1995. Random Walks. Random Walks and Random Environments, vol. 1. Clarendon, Oxford.
Hughes, B.D., 1996. Random Environments. Random Walks and Random Environments, vol. 2. Clarendon, Oxford.
Hugues, S., Fetler, L., Bonifaz, L., Helft, J., Amblard, F., Amigorena, S., 2004. Distinct T cell dynamics in lymph nodes during the induction of tolerance and immunity. Nat. Immunol. 5(12), 1235–1242.
Kulesa, P., Ellies, D.L., Trainor, P.A., 2004. Comparative analysis of neural crest cell death, migration, and function during vertebrate embryogenesis. Dev. Dyn. 229(1), 14–29.
Kusumi, A., Sako, Y., 1996. Cell surface organization by the membrane skeleton. Curr. Opin. Cell Biol. 8(4), 566–574.
Lamb, R.F., Ozanne, B.W., Roy, C., McGarry, L., Stipp, C., Mangeat, P., Jay, D.G., 1997. Essential functions of ezrin in maintenance of cell shape and lamellipodium extension in normal and transformed fibroblasts. Curr. Biol. 7(9), 682–688.
Luby-Phelps, K., 2000. Cytoarchitecture and physical properties of cytoplasm: volume, viscosity, diffusion, intracellular surface area. Int. Rev. Cytol. 192, 189–221.
Maly, I.V., Vorobjev, I.A., 2002. Centrosome-dependent anisotropic random walk of cytoplasmic vesicles. Cell Biol. Int. 26(9), 791–799.
Marshall, W.F., Marko, J.F., Agard, D.A., Sedat, J.W., 2001. Chromosome elasticity and mitotic polar ejection force measured in living Drosophila embryos by four-dimensional microscopy-based motion analysis. Curr. Biol. 11(8), 569–578.
Martin, D.S., Forstner, M.B., Kas, J.A., 2002. Apparent subdiffusion inherent to single particle tracking. Biophys. J. 83(4), 2109–2117.
Niggemann, B., Drell, T.L. 4th, Joseph, J., Weidt, C., Lang, K., Zaenker, K.S., Entschladen, F., 2004. Tumor cell locomotion: differential dynamics of spontaneous and induced migration in a 3D collagen matrix. Exp. Cell Res. 298(1), 178–187.
Ochi, M.K., 1990. Applied Probability and Stochastic Processes. Wiley, New York.
Ordemann, A., Berkolaiko, G., Havlin, S., Bunde, A., 2000. Swelling-collapse transition of self-attracting walks. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 61(2), R1005–R1007.
Ott, A., Bouchaud, J.P., Langevin, D., Urbach, W., 1990. Anomalous diffusion in “living polymers”: a genuine Levy flight? Phys. Rev. Lett. 65(17), 2201–2204.
Peng, C.K., Buldyrev, S.V., Goldberger, A.L., Havlin, S., Sciortino, F., Simons, M., Stanley, H.E., 1992. Long-range correlations in nucleotide sequences. Nature 356(6365), 168–170.
Rabinovich, S., Roman, H.E., Havlin, S., Bunde, A., 1996. Critical dimensions for random walks on random-walk chains. Phys. Rev. E. Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 54(4), 3606–3608.
Rief, M., Rock, R.S., Mehta, A.D., Mooseker, M.S., Cheney, R.E., Spudich, J.A., 2000. Myosin-V stepping kinetics: a molecular model for processivity. Proc. Natl. Acad. Sci. USA 97(17), 9482–9486.
Rock, R.S., Rice, S.E., Wells, A.L., Purcell, T.J., Spudich, J.A., Sweeney, H.L., 2001. Myosin VI is a processive motor with a large step size. Proc. Natl. Acad. Sci. USA 98(24), 13655–13659.
Rudnick, J., Hu, Y., 1988. Winding angle of a self-avoiding random walk. Phys. Rev. Lett. 60(8), 712–715.
Salomé, L., Cazeils, J.L., Lopez, A., Tocanne, J.F., 1998. Characterization of membrane domains by FRAP experiments at variable observation areas. Eur. Biophys. J. 27(4), 391–402.
Savino, T.M., Gebrane-Younes, J., De Mey, J., Sibarita, J.B., Hernandez-Verdun, D., 2001. Nucleolar assembly of the RNA processing machinery in living cells. J. Cell Biol. 153(5), 1097–1110.
Saxton, M.J., 1993. Lateral diffusion in an archipelago. Single-particle diffusion. Biophys. J. 64(6), 1766–1780.
Saxton, M.J., 1994. Single-particle tracking: models of directed transport. Biophys. J. 67(5), 2110–2119.
Saxton, M.J., 1995. Single-particle tracking: effects of corrals. Biophys. J. 69(2), 389–398.
Saxton, M.J., 1997. Single-particle tracking: the distribution of diffusion coefficients. Biophys. J. 72(4), 1744–1753.
Sciutto, S.J., 1995. Study of the shape of random walks: II. Inertia moment ratios and the two-dimensional asphericity. J. Phys. A: Math. Gen. 28, 3667–3679.
Shaevitz, J.W., Abbondanzieri, E.A., Landick, R., Block, S.M., 2003. Backtracking by single RNA polymerase molecules observed at near-base-pair resolution. Nature 426(6967), 684–687.
Sheetz, M.P., 1999. Motor and cargo interactions. Eur. J. Biochem. 262(1), 19–25.
Simson, R., Sheets, E.D., Jacobson, K., 1995. Detection of temporary lateral confinement of membrane proteins using single-particle tracking analysis. Biophys. J. 69, 989–993.
Spudich, J.A., 2001. The myosin swinging cross-bridge model. Nat. Rev. Mol. Cell Biol. 2(5), 387–392.
Tsien, R.Y., 1998. The green fluorescent protein. Annu. Rev. Biochem. 67, 509–544.
Ulrich, F., Concha, M.L., Heid, P.J., Voss, E., Witzel, S., Roehl, H., Tada, M., Wilson, S.W., Adams, R.J., Soll, D.R., Heisenberg, C.P., 2003. Slb/Wnt11 controls hypoblast cell migration and morphogenesis at the onset of zebrafish gastrulation. Development 130(22), 5375–5384.
Waharte, F., Brown, C.M., Coscoy, S., Coudrier, E., Amblard, F., 2005. A two-photon FRAP analysis of the cytoskeleton dynamics in the microvilli of intestinal cells. Biophys. J. 88(2), 1467–1478.
Wei, G., 1995. Exact shapes of random walks in two dimensions. Physica A 222, 152–154.
Wong, I.Y., Gardel, M.L., Reichman, D.R., Weeks, E.R., Valentine, M.T., Bausch, A.R., Weitz, D.A., 2004. Anomalous diffusion probes microstructure dynamics of entangled F-actin networks. Phys. Rev. Lett. 92(17), 178101.
Zhang, F., Lee, G.M., Jacobson, K., 1993. Protein lateral mobility as a reflection of membrane microstructure. Bioessays 15(9), 579–588.
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Coscoy, S., Huguet, E. & Amblard, F. Statistical Analysis of Sets of Random Walks: How to Resolve Their Generating Mechanism. Bull. Math. Biol. 69, 2467–2492 (2007). https://doi.org/10.1007/s11538-007-9227-8
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DOI: https://doi.org/10.1007/s11538-007-9227-8